### 3.1505 $$\int \frac{(d+e x)^5}{a^2+2 a b x+b^2 x^2} \, dx$$

Optimal. Leaf size=131 $\frac{5 e^4 (a+b x)^3 (b d-a e)}{3 b^6}+\frac{5 e^3 (a+b x)^2 (b d-a e)^2}{b^6}+\frac{10 e^2 x (b d-a e)^3}{b^5}-\frac{(b d-a e)^5}{b^6 (a+b x)}+\frac{5 e (b d-a e)^4 \log (a+b x)}{b^6}+\frac{e^5 (a+b x)^4}{4 b^6}$

[Out]

(10*e^2*(b*d - a*e)^3*x)/b^5 - (b*d - a*e)^5/(b^6*(a + b*x)) + (5*e^3*(b*d - a*e)^2*(a + b*x)^2)/b^6 + (5*e^4*
(b*d - a*e)*(a + b*x)^3)/(3*b^6) + (e^5*(a + b*x)^4)/(4*b^6) + (5*e*(b*d - a*e)^4*Log[a + b*x])/b^6

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Rubi [A]  time = 0.146367, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.077, Rules used = {27, 43} $\frac{5 e^4 (a+b x)^3 (b d-a e)}{3 b^6}+\frac{5 e^3 (a+b x)^2 (b d-a e)^2}{b^6}+\frac{10 e^2 x (b d-a e)^3}{b^5}-\frac{(b d-a e)^5}{b^6 (a+b x)}+\frac{5 e (b d-a e)^4 \log (a+b x)}{b^6}+\frac{e^5 (a+b x)^4}{4 b^6}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(10*e^2*(b*d - a*e)^3*x)/b^5 - (b*d - a*e)^5/(b^6*(a + b*x)) + (5*e^3*(b*d - a*e)^2*(a + b*x)^2)/b^6 + (5*e^4*
(b*d - a*e)*(a + b*x)^3)/(3*b^6) + (e^5*(a + b*x)^4)/(4*b^6) + (5*e*(b*d - a*e)^4*Log[a + b*x])/b^6

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(d+e x)^5}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac{(d+e x)^5}{(a+b x)^2} \, dx\\ &=\int \left (\frac{10 e^2 (b d-a e)^3}{b^5}+\frac{(b d-a e)^5}{b^5 (a+b x)^2}+\frac{5 e (b d-a e)^4}{b^5 (a+b x)}+\frac{10 e^3 (b d-a e)^2 (a+b x)}{b^5}+\frac{5 e^4 (b d-a e) (a+b x)^2}{b^5}+\frac{e^5 (a+b x)^3}{b^5}\right ) \, dx\\ &=\frac{10 e^2 (b d-a e)^3 x}{b^5}-\frac{(b d-a e)^5}{b^6 (a+b x)}+\frac{5 e^3 (b d-a e)^2 (a+b x)^2}{b^6}+\frac{5 e^4 (b d-a e) (a+b x)^3}{3 b^6}+\frac{e^5 (a+b x)^4}{4 b^6}+\frac{5 e (b d-a e)^4 \log (a+b x)}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.0783108, size = 230, normalized size = 1.76 $\frac{10 a^2 b^3 e^2 \left (-24 d^2 e x-12 d^3+12 d e^2 x^2+e^3 x^3\right )+30 a^3 b^2 e^3 \left (4 d^2+6 d e x-e^2 x^2\right )-12 a^4 b e^4 (5 d+4 e x)+12 a^5 e^5-5 a b^4 e \left (36 d^2 e^2 x^2-24 d^3 e x-12 d^4+8 d e^3 x^3+e^4 x^4\right )+60 e (a+b x) (b d-a e)^4 \log (a+b x)+b^5 \left (120 d^3 e^2 x^2+60 d^2 e^3 x^3-12 d^5+20 d e^4 x^4+3 e^5 x^5\right )}{12 b^6 (a+b x)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(12*a^5*e^5 - 12*a^4*b*e^4*(5*d + 4*e*x) + 30*a^3*b^2*e^3*(4*d^2 + 6*d*e*x - e^2*x^2) + 10*a^2*b^3*e^2*(-12*d^
3 - 24*d^2*e*x + 12*d*e^2*x^2 + e^3*x^3) - 5*a*b^4*e*(-12*d^4 - 24*d^3*e*x + 36*d^2*e^2*x^2 + 8*d*e^3*x^3 + e^
4*x^4) + b^5*(-12*d^5 + 120*d^3*e^2*x^2 + 60*d^2*e^3*x^3 + 20*d*e^4*x^4 + 3*e^5*x^5) + 60*e*(b*d - a*e)^4*(a +
b*x)*Log[a + b*x])/(12*b^6*(a + b*x))

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Maple [B]  time = 0.048, size = 326, normalized size = 2.5 \begin{align*}{\frac{{e}^{5}{x}^{4}}{4\,{b}^{2}}}-{\frac{2\,{e}^{5}{x}^{3}a}{3\,{b}^{3}}}+{\frac{5\,{e}^{4}{x}^{3}d}{3\,{b}^{2}}}+{\frac{3\,{e}^{5}{x}^{2}{a}^{2}}{2\,{b}^{4}}}-5\,{\frac{{e}^{4}{x}^{2}ad}{{b}^{3}}}+5\,{\frac{{e}^{3}{x}^{2}{d}^{2}}{{b}^{2}}}-4\,{\frac{{a}^{3}{e}^{5}x}{{b}^{5}}}+15\,{\frac{{a}^{2}d{e}^{4}x}{{b}^{4}}}-20\,{\frac{a{d}^{2}{e}^{3}x}{{b}^{3}}}+10\,{\frac{{d}^{3}{e}^{2}x}{{b}^{2}}}+5\,{\frac{{e}^{5}\ln \left ( bx+a \right ){a}^{4}}{{b}^{6}}}-20\,{\frac{{e}^{4}\ln \left ( bx+a \right ){a}^{3}d}{{b}^{5}}}+30\,{\frac{{e}^{3}\ln \left ( bx+a \right ){d}^{2}{a}^{2}}{{b}^{4}}}-20\,{\frac{{e}^{2}\ln \left ( bx+a \right ) a{d}^{3}}{{b}^{3}}}+5\,{\frac{e\ln \left ( bx+a \right ){d}^{4}}{{b}^{2}}}+{\frac{{a}^{5}{e}^{5}}{{b}^{6} \left ( bx+a \right ) }}-5\,{\frac{{a}^{4}d{e}^{4}}{{b}^{5} \left ( bx+a \right ) }}+10\,{\frac{{a}^{3}{d}^{2}{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}-10\,{\frac{{a}^{2}{d}^{3}{e}^{2}}{{b}^{3} \left ( bx+a \right ) }}+5\,{\frac{a{d}^{4}e}{{b}^{2} \left ( bx+a \right ) }}-{\frac{{d}^{5}}{b \left ( bx+a \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2),x)

[Out]

1/4*e^5/b^2*x^4-2/3*e^5/b^3*x^3*a+5/3*e^4/b^2*x^3*d+3/2*e^5/b^4*x^2*a^2-5*e^4/b^3*x^2*a*d+5*e^3/b^2*x^2*d^2-4*
e^5/b^5*a^3*x+15*e^4/b^4*a^2*d*x-20*e^3/b^3*a*d^2*x+10*e^2/b^2*d^3*x+5/b^6*e^5*ln(b*x+a)*a^4-20/b^5*e^4*ln(b*x
+a)*a^3*d+30/b^4*e^3*ln(b*x+a)*d^2*a^2-20/b^3*e^2*ln(b*x+a)*a*d^3+5/b^2*e*ln(b*x+a)*d^4+1/b^6/(b*x+a)*a^5*e^5-
5/b^5/(b*x+a)*a^4*d*e^4+10/b^4/(b*x+a)*a^3*d^2*e^3-10/b^3/(b*x+a)*a^2*d^3*e^2+5/b^2/(b*x+a)*a*d^4*e-1/b/(b*x+a
)*d^5

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Maxima [B]  time = 1.1607, size = 358, normalized size = 2.73 \begin{align*} -\frac{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}}{b^{7} x + a b^{6}} + \frac{3 \, b^{3} e^{5} x^{4} + 4 \,{\left (5 \, b^{3} d e^{4} - 2 \, a b^{2} e^{5}\right )} x^{3} + 6 \,{\left (10 \, b^{3} d^{2} e^{3} - 10 \, a b^{2} d e^{4} + 3 \, a^{2} b e^{5}\right )} x^{2} + 12 \,{\left (10 \, b^{3} d^{3} e^{2} - 20 \, a b^{2} d^{2} e^{3} + 15 \, a^{2} b d e^{4} - 4 \, a^{3} e^{5}\right )} x}{12 \, b^{5}} + \frac{5 \,{\left (b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} \log \left (b x + a\right )}{b^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

-(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)/(b^7*x + a*b^6)
+ 1/12*(3*b^3*e^5*x^4 + 4*(5*b^3*d*e^4 - 2*a*b^2*e^5)*x^3 + 6*(10*b^3*d^2*e^3 - 10*a*b^2*d*e^4 + 3*a^2*b*e^5)
*x^2 + 12*(10*b^3*d^3*e^2 - 20*a*b^2*d^2*e^3 + 15*a^2*b*d*e^4 - 4*a^3*e^5)*x)/b^5 + 5*(b^4*d^4*e - 4*a*b^3*d^3
*e^2 + 6*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4 + a^4*e^5)*log(b*x + a)/b^6

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Fricas [B]  time = 1.81274, size = 767, normalized size = 5.85 \begin{align*} \frac{3 \, b^{5} e^{5} x^{5} - 12 \, b^{5} d^{5} + 60 \, a b^{4} d^{4} e - 120 \, a^{2} b^{3} d^{3} e^{2} + 120 \, a^{3} b^{2} d^{2} e^{3} - 60 \, a^{4} b d e^{4} + 12 \, a^{5} e^{5} + 5 \,{\left (4 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 10 \,{\left (6 \, b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 30 \,{\left (4 \, b^{5} d^{3} e^{2} - 6 \, a b^{4} d^{2} e^{3} + 4 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 12 \,{\left (10 \, a b^{4} d^{3} e^{2} - 20 \, a^{2} b^{3} d^{2} e^{3} + 15 \, a^{3} b^{2} d e^{4} - 4 \, a^{4} b e^{5}\right )} x + 60 \,{\left (a b^{4} d^{4} e - 4 \, a^{2} b^{3} d^{3} e^{2} + 6 \, a^{3} b^{2} d^{2} e^{3} - 4 \, a^{4} b d e^{4} + a^{5} e^{5} +{\left (b^{5} d^{4} e - 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (b x + a\right )}{12 \,{\left (b^{7} x + a b^{6}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

1/12*(3*b^5*e^5*x^5 - 12*b^5*d^5 + 60*a*b^4*d^4*e - 120*a^2*b^3*d^3*e^2 + 120*a^3*b^2*d^2*e^3 - 60*a^4*b*d*e^4
+ 12*a^5*e^5 + 5*(4*b^5*d*e^4 - a*b^4*e^5)*x^4 + 10*(6*b^5*d^2*e^3 - 4*a*b^4*d*e^4 + a^2*b^3*e^5)*x^3 + 30*(4
*b^5*d^3*e^2 - 6*a*b^4*d^2*e^3 + 4*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 12*(10*a*b^4*d^3*e^2 - 20*a^2*b^3*d^2*e^
3 + 15*a^3*b^2*d*e^4 - 4*a^4*b*e^5)*x + 60*(a*b^4*d^4*e - 4*a^2*b^3*d^3*e^2 + 6*a^3*b^2*d^2*e^3 - 4*a^4*b*d*e^
4 + a^5*e^5 + (b^5*d^4*e - 4*a*b^4*d^3*e^2 + 6*a^2*b^3*d^2*e^3 - 4*a^3*b^2*d*e^4 + a^4*b*e^5)*x)*log(b*x + a))
/(b^7*x + a*b^6)

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Sympy [A]  time = 1.20108, size = 224, normalized size = 1.71 \begin{align*} \frac{a^{5} e^{5} - 5 a^{4} b d e^{4} + 10 a^{3} b^{2} d^{2} e^{3} - 10 a^{2} b^{3} d^{3} e^{2} + 5 a b^{4} d^{4} e - b^{5} d^{5}}{a b^{6} + b^{7} x} + \frac{e^{5} x^{4}}{4 b^{2}} - \frac{x^{3} \left (2 a e^{5} - 5 b d e^{4}\right )}{3 b^{3}} + \frac{x^{2} \left (3 a^{2} e^{5} - 10 a b d e^{4} + 10 b^{2} d^{2} e^{3}\right )}{2 b^{4}} - \frac{x \left (4 a^{3} e^{5} - 15 a^{2} b d e^{4} + 20 a b^{2} d^{2} e^{3} - 10 b^{3} d^{3} e^{2}\right )}{b^{5}} + \frac{5 e \left (a e - b d\right )^{4} \log{\left (a + b x \right )}}{b^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

(a**5*e**5 - 5*a**4*b*d*e**4 + 10*a**3*b**2*d**2*e**3 - 10*a**2*b**3*d**3*e**2 + 5*a*b**4*d**4*e - b**5*d**5)/
(a*b**6 + b**7*x) + e**5*x**4/(4*b**2) - x**3*(2*a*e**5 - 5*b*d*e**4)/(3*b**3) + x**2*(3*a**2*e**5 - 10*a*b*d*
e**4 + 10*b**2*d**2*e**3)/(2*b**4) - x*(4*a**3*e**5 - 15*a**2*b*d*e**4 + 20*a*b**2*d**2*e**3 - 10*b**3*d**3*e*
*2)/b**5 + 5*e*(a*e - b*d)**4*log(a + b*x)/b**6

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Giac [B]  time = 1.15016, size = 347, normalized size = 2.65 \begin{align*} \frac{5 \,{\left (b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}}{{\left (b x + a\right )} b^{6}} + \frac{3 \, b^{6} x^{4} e^{5} + 20 \, b^{6} d x^{3} e^{4} + 60 \, b^{6} d^{2} x^{2} e^{3} + 120 \, b^{6} d^{3} x e^{2} - 8 \, a b^{5} x^{3} e^{5} - 60 \, a b^{5} d x^{2} e^{4} - 240 \, a b^{5} d^{2} x e^{3} + 18 \, a^{2} b^{4} x^{2} e^{5} + 180 \, a^{2} b^{4} d x e^{4} - 48 \, a^{3} b^{3} x e^{5}}{12 \, b^{8}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

5*(b^4*d^4*e - 4*a*b^3*d^3*e^2 + 6*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4 + a^4*e^5)*log(abs(b*x + a))/b^6 - (b^5*d^5
- 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)/((b*x + a)*b^6) + 1/12*(
3*b^6*x^4*e^5 + 20*b^6*d*x^3*e^4 + 60*b^6*d^2*x^2*e^3 + 120*b^6*d^3*x*e^2 - 8*a*b^5*x^3*e^5 - 60*a*b^5*d*x^2*e
^4 - 240*a*b^5*d^2*x*e^3 + 18*a^2*b^4*x^2*e^5 + 180*a^2*b^4*d*x*e^4 - 48*a^3*b^3*x*e^5)/b^8